The product of combinations, and how fast it grows

Question

What is the nature of $\prod_{k=1}^{k=n}\binom{n}{k}$ as a function of n?

How fast does it grow in comparison with $2^n$, etc?

My intuition is that:

$$\prod_{k=1}^{k=n}\binom{n}{k}$$

Grows much faster than

$$2^n$$

Simply because the summation is $2^n$. How fast does it grow in comparison, though?

Answer 1

This sequence is OEIS A001142. The entry gives the asymptotic approximation

$$\frac{A^2e^{\frac{n^2}2+n-\frac{1}{12}}}{n^{\frac{n}2+\frac{1}2}(2\pi)^{\frac{n+1}2}}\;,$$

where $A\approx 1.282427129$ is the Glaisher-Kinkelin constant.

Answer 2

Hint

If you only look at the "middle term" $\binom{2n}{n}$ for the even case and you use Stirling's approximation, you'll find that just this term is growing in the range of $$\frac{2^{2n}}{\sqrt{ \pi n}}$$